Awareness programs control infectious disease – Multiple delay induced mathematical model

Abstract

We propose and analyze a mathematical model to study the impact of awareness programs on an infectious disease outbreak. These programs induce behavioral changes in the population, which divide the susceptible class into two subclasses, aware susceptible and unaware susceptible. The system can have a disease-free equilibrium and an endemic equilibrium. The expression of the basic reproduction number and the conditions for the stability of the equilibria are derived. We further improve and study the model by introducing two time-delay factors, one for the time lag in memory fading of aware people and one for the delay between cases of disease occurring and mounting awareness programs. The delayed system has positive bounded solutions. We study various cases for the time delays and show that in general the system develops limit cycle oscillation through a Hopf bifurcation for increasing time delays. We show that under certain conditions on the parameters, the system is permanent. To verify our analytical findings, the numerical simulations on the model, using realistic parameters for Pneumococcus are performed.